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Empirical Spectral Distribution of a Matrix Under Perturbation

  • Florent Benaych-Georges
  • Nathanaël EnriquezEmail author
  • Alkéos Michaïl
Article

Abstract

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.

Keywords

Random matrices Perturbation theory Wigner matrices Band matrices Hilbert transform Spectral density 

Mathematics Subject Classification (2010)

15A52 60B20 47A55 46L54 

Notes

Acknowledgements

We thank Jean-Philippe Bouchaud, Guy David and Vincent Vargas for some fruitful discussions. We are also glad to thank the GDR MEGA for partial support.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Université Paris DescartesParis Cedex 06France
  2. 2.Laboratoire de mathématiques d’Orsay (UMR 8628)Université Paris-SudOrsayFrance

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